# Spontaneous Symmetry Breaking

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• Path Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics Dennis V. Perepelitsa MIT Department of Physics 70 Amherst Ave. Cambridge, MA 02142 Abstract We present the path integral formulation of quantum mechanics and demon- strate its equivalence to the Schr¨odinger picture. We apply the method to the free particle and quantum harmonic oscillator, investigate the Euclidean path integral, and discuss other applications. 1 Introduction A fundamental question in quantum mechanics is how does the state of a particle evolve with time? That is, the determination the time-evolution ψ(t) of some initial | i state ψ(t ) . Quantum mechanics is fully predictive [3] in the sense that initial | 0 i conditions and knowledge of the potential occupied by the particle is enough to fully specify the state of the particle for all future times.1 In the early twentieth century, Erwin Schr¨odinger derived an equation speciﬁes how the instantaneous change in the wavefunction d ψ(t) depends on the system dt | i inhabited by the state in the form of the Hamiltonian. In this formulation, the eigenstates of the Hamiltonian play an important role, since their time-evolution is easy to calculate (i.e. they are stationary). A well-established method of solution, after the entire eigenspectrum of Hˆ is known, is to decompose the initial state into this eigenbasis, apply time evolution to each and then reassemble the eigenstates. That is, 1In the analysis below, we consider only the position of a particle, and not any other quantum property such as spin. 2 D.V. Perepelitsa n=∞ ψ(t) = exp [ iE t/~] n ψ(t ) n (1) | i − n h | 0 i| i n=0 X This (Hamiltonian) formulation works in many cases.
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• An Introduction to Quantum Field Theory
AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ...........................
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• Quantum Field Theory*
Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement.
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• Richard Lieu
Richard Lieu E-mail address: [email protected] Academic Qualiﬁcations: Oct., 1975 - June, 1978: BSc (1st class honors), Imperial College London, Major in Physics. Oct., 1978 - June, 1981: DIC, PhD, Imperial College London. Thesis title: Radiation Transport within the source of Hard Cosmic X-ray Photons. Career Record 21 June, 2008 - present: 10th Distinguished Professor and Chair of the Department of Physics, University of Alabama in Huntsville. 14 Sept., 2013 - 21 Jan., 2015, Chair of the Department of Physics, Univer- sity of Alabama in Huntsville. 20 Aug., 2004 - 20 June, 2008: Professor at the Department of Physics, University of Alabama, Huntsville. 16 Aug., 1999 - present: Associate Professor at the Department of Physics, University of Alabama, Huntsville. 16 Aug., 1995 - 15 Aug., 1999: Assistant Professor at the Department of Physics, University of Alabama, Huntsville. 1 1 Nov., 1991 - 15 Aug., 1995: Assistant Research Astronomer at the Center for EUV Astrophysics, University of California, Berkeley. 1 Aug., 1985 - 31 Oct., 1991: Research Assistant at the Astrophysics Group, Blackett Laboratory, Imperial College, London, England. 1 Aug., 1981 - 30 June, 1985: Postdoctoral Research Fellow and Sessional Instructor at the Dept. of Physics and Astronomy, and at the Dept. of Electrical Engineering, University of Calgary, Alberta, Canada. Awards and Prizes UAH Foundation Researcher of the year award, April 2007. Sigma Xi Huntsville chapter `researcher of the year' award, with Lloyd W. Hillman, April 2005. Three times `Discovery of the year award', 1995, 1994, and 1993, Center for EUV Astrophysics, UC Berkeley. `Outstanding software development of the year award 1993', with James W. Lewis, Center for EUV Astrophysics, UC Berkeley.
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• On the Definition of the Renormalization Constants in Quantum Electrodynamics
On the Definition of the Renormalization Constants in Quantum Electrodynamics Autor(en): Källén, Gunnar Objekttyp: Article Zeitschrift: Helvetica Physica Acta Band(Jahr): 25(1952) Heft IV Erstellt am: Mar 18, 2014 Persistenter Link: http://dx.doi.org/10.5169/seals-112316 Nutzungsbedingungen Mit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert. Die angebotenen Dokumente stehen für nicht-kommerzielle Zwecke in Lehre, Forschung und für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und unter deren Einhaltung weitergegeben werden. Die Speicherung von Teilen des elektronischen Angebots auf anderen Servern ist nur mit vorheriger schriftlicher Genehmigung möglich. Die Rechte für diese und andere Nutzungsarten der Inhalte liegen beim Herausgeber bzw. beim Verlag. Ein Dienst der ETH-Bibliothek Rämistrasse 101, 8092 Zürich, Schweiz [email protected] http://retro.seals.ch On the Definition of the Renormalization Constants in Quantum Electrodynamics by Gunnar Källen.*) Swiss Federal Institute of Technology, Zürich. (14.11.1952.) Summary. A formulation of quantum electrodynamics in terms of the renor- malized Heisenberg operators and the experimental mass and charge of the electron is given. The renormalization constants are implicitly defined and ex¬ pressed as integrals over finite functions in momentum space. No discussion of the convergence of these integrals or of the existence of rigorous solutions is given. Introduction. The renormalization method in quantum electrodynamics has been investigated by many authors, and it has been proved by Dyson1) that every term in a formal expansion in powers of the coupling constant of various expressions is a finite quantity.
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• THE DEVELOPMENT of the SPACE-TIME VIEW of QUANTUM ELECTRODYNAMICS∗ by Richard P
THE DEVELOPMENT OF THE SPACE-TIME VIEW OF QUANTUM ELECTRODYNAMICS∗ by Richard P. Feynman California Institute of Technology, Pasadena, California Nobel Lecture, December 11, 1965. We have a habit in writing articles published in scientiﬁc journals to make the work as ﬁnished as possible, to cover all the tracks, to not worry about the blind alleys or to describe how you had the wrong idea ﬁrst, and so on. So there isn’t any place to publish, in a digniﬁed manner, what you actually did in order to get to do the work, although, there has been in these days, some interest in this kind of thing. Since winning the prize is a personal thing, I thought I could be excused in this particular situation, if I were to talk personally about my relationship to quantum electrodynamics, rather than to discuss the subject itself in a reﬁned and ﬁnished fashion. Furthermore, since there are three people who have won the prize in physics, if they are all going to be talking about quantum electrodynamics itself, one might become bored with the subject. So, what I would like to tell you about today are the sequence of events, really the sequence of ideas, which occurred, and by which I ﬁnally came out the other end with an unsolved problem for which I ultimately received a prize. I realize that a truly scientiﬁc paper would be of greater value, but such a paper I could publish in regular journals. So, I shall use this Nobel Lecture as an opportunity to do something of less value, but which I cannot do elsewhere.
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• Annual Report 2004
ANNUAL REPORT 2004 HELSINKI UNIVERSITY OF TECHNOLOGY Low Temperature Laboratory Brain Research Unit and Low Temperature Physics Research http://boojum.hut.fi - 2 - PREFACE...............................................................................................................5 SCIENTIFIC ADVISORY BOARD .......................................................................7 PERSONNEL .........................................................................................................7 SENIOR RESEARCHERS..................................................................................7 ADMINISTRATION AND TECHNICAL PERSONNEL...................................8 GRADUATE STUDENTS (SUPERVISOR).......................................................8 UNDERGRADUATE STUDENTS.....................................................................9 VISITORS FOR EU PROJECTS ......................................................................10 OTHER VISITORS ..........................................................................................11 GROUP VISITS................................................................................................13 OLLI V. LOUNASMAA MEMORIAL PRIZE 2004 ............................................15 INTERNATIONAL COLLABORATIONS ..........................................................16 CERN COLLABORATION (COMPASS) ........................................................16 COSLAB (COSMOLOGY IN THE LABORATORY)......................................16 ULTI III - ULTRA LOW TEMPERATURE INSTALLATION
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• (Aka Second Quantization) 1 Quantum Field Theory
221B Lecture Notes Quantum Field Theory (a.k.a. Second Quantization) 1 Quantum Field Theory Why quantum ﬁeld theory? We know quantum mechanics works perfectly well for many systems we had looked at already. Then why go to a new formalism? The following few sections describe motivation for the quantum ﬁeld theory, which I introduce as a re-formulation of multi-body quantum mechanics with identical physics content. 1.1 Limitations of Multi-body Schr¨odinger Wave Func- tion We used totally anti-symmetrized Slater determinants for the study of atoms, molecules, nuclei. Already with the number of particles in these systems, say, about 100, the use of multi-body wave function is quite cumbersome. Mention a wave function of an Avogardro number of particles! Not only it is completely impractical to talk about a wave function with 6 × 1023 coordinates for each particle, we even do not know if it is supposed to have 6 × 1023 or 6 × 1023 + 1 coordinates, and the property of the system of our interest shouldn’t be concerned with such a tiny (?) diﬀerence. Another limitation of the multi-body wave functions is that it is incapable of describing processes where the number of particles changes. For instance, think about the emission of a photon from the excited state of an atom. The wave function would contain coordinates for the electrons in the atom and the nucleus in the initial state. The ﬁnal state contains yet another particle, photon in this case. But the Schr¨odinger equation is a diﬀerential equation acting on the arguments of the Schr¨odingerwave function, and can never change the number of arguments.
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• TASI 2008 Lectures: Introduction to Supersymmetry And
TASI 2008 Lectures: Introduction to Supersymmetry and Supersymmetry Breaking Yuri Shirman Department of Physics and Astronomy University of California, Irvine, CA 92697. [email protected] Abstract These lectures, presented at TASI 08 school, provide an introduction to supersymmetry and supersymmetry breaking. We present basic formalism of supersymmetry, super- symmetric non-renormalization theorems, and summarize non-perturbative dynamics of supersymmetric QCD. We then turn to discussion of tree level, non-perturbative, and metastable supersymmetry breaking. We introduce Minimal Supersymmetric Standard Model and discuss soft parameters in the Lagrangian. Finally we discuss several mech- anisms for communicating the supersymmetry breaking between the hidden and visible sectors. arXiv:0907.0039v1 [hep-ph] 1 Jul 2009 Contents 1 Introduction 2 1.1 Motivation..................................... 2 1.2 Weylfermions................................... 4 1.3 Aﬁrstlookatsupersymmetry . .. 5 2 Constructing supersymmetric Lagrangians 6 2.1 Wess-ZuminoModel ............................... 6 2.2 Superﬁeldformalism .............................. 8 2.3 VectorSuperﬁeld ................................. 12 2.4 Supersymmetric U(1)gaugetheory ....................... 13 2.5 Non-abeliangaugetheory . .. 15 3 Non-renormalization theorems 16 3.1 R-symmetry.................................... 17 3.2 Superpotentialterms . .. .. .. 17 3.3 Gaugecouplingrenormalization . ..... 19 3.4 D-termrenormalization. ... 20 4 Non-perturbative dynamics in SUSY QCD 20 4.1 Aﬄeck-Dine-Seiberg
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• Nobelprisen I Fysikk 2013
Nr. 4 – 2013 75. årgang Fra Utgiver: Norsk Fysisk Selskap Redaktører: Øyvind Grøn Emil J. Samuelsen Fysik ke ns Redaksjonssekretær: Karl Måseide Innhold Ould-Saada, Raklev og Read: Nobelprisen i fysikk 2013 . 101 Verden Øyvind Grøn: Romelven og treg draeffekt . 104 Emil J. Samuelsen: Friksjon og gli . 111 Frå Redaktørane . 98 In Memoriam . 98 Øystein H. Fischer FFV Gratulerer . 100 Per Chr. Hemmer 80 år Hva skjer . 116 Birkelandforelesningen 2013 Yaras Birkelandpris 2014 Konferanse om kvinner i fysikk Bokomtale . 118 Arnt Inge Vistnes: Svingninger og bølger Trim i FFV . 118 Nye Doktorer . 119 PhD Marit Sandstad Nobelprisvinnerne i fysikk 2013: François Englert (t.v.) og Peter Higgs (Se artikkel om Nobelprisen i fysikk) ISSN-0015-9247 SIDE 98 FRA FYSIKKENS VERDEN 4/13 S ve n O lu f S ør en se n FRA FYSIKKENS VERDEN 4/13 SIDE 99 supraledning, hvor han og hans team gjorde flere fysikkmiljøet i Geneve til Trondheim. Øystein banebrytende oppdagelser av nye fenomen og egen­ takket jevnlig ja til å delta på vitenskapelige kon­ skaper som har bidratt til å øke vår forståelse av feranser, workshops og seminar her i Norge, der han disse komplekse materialene. Han er særlig kjent alltid bidro til så vel høy kvalitet som god stemning. for sine studier av magnetiske ternære supraledere På den forskningpolitiske arena ledet han i 1986 den og komplekse oksid supraledere med høy "kri­ første fysikkevalueringen i regi av det daværende tisk" temperatur (dvs. høy-T c supraledere). Av Norges almenvitenskapelige forskningsråd (NAVF), hans arbeider kan nevnes påvisningen av koeksis­ den såkalte Fischer-komiteen.
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• Weak Interaction Processes: Which Quantum Information Is Revealed?
Weak Interaction Processes: Which Quantum Information is revealed? B. C. Hiesmayr1 1University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Vienna, Austria We analyze the achievable limits of the quantum information processing of the weak interaction revealed by hyperons with spin. We ﬁnd that the weak decay process corresponds to an interferometric device with a ﬁxed visibility and ﬁxed phase diﬀerence for each hyperon. Nature chooses rather low visibilities expressing a preference to parity conserving or violating processes (except for the decay Σ+ pπ0). The decay process can be considered as an open quantum channel that carries the information of the−→ hyperon spin to the angular distribution of the momentum of the daughter particles. We ﬁnd a simple geometrical information theoretic 1 α interpretation of this process: two quantization axes are chosen spontaneously with probabilities ±2 where α is proportional to the visibility times the real part of the phase shift. Diﬀerently stated the weak interaction process corresponds to spin measurements with an imperfect Stern-Gerlach apparatus. Equipped with this information theoretic insight we show how entanglement can be measured in these systems and why Bell’s nonlocality (in contradiction to common misconception in literature) cannot be revealed in hyperon decays. We study also under which circumstances contextuality can be revealed. PACS numbers: 03.67.-a, 14.20.Jn, 03.65.Ud Weak interactions are one out of the four fundamental in- systems. teractions that we think that rules our universe. The weak in- Information theoretic content of an interfering and de- teraction is the only interaction that breaks the parity symme- caying system: Let us start by assuming that the initial state try and the combined charge-conjugation–parity( P) symme- of a decaying hyperon is in a separable state between momen- C try.
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• 1 Drawing Feynman Diagrams
1 Drawing Feynman Diagrams 1. A fermion (quark, lepton, neutrino) is drawn by a straight line with an arrow pointing to the left: f f 2. An antifermion is drawn by a straight line with an arrow pointing to the right: f f 3. A photon or W ±, Z0 boson is drawn by a wavy line: γ W ±;Z0 4. A gluon is drawn by a curled line: g 5. The emission of a photon from a lepton or a quark doesn’t change the fermion: γ l; q l; q But a photon cannot be emitted from a neutrino: γ ν ν 6. The emission of a W ± from a fermion changes the ﬂavour of the fermion in the following way: − − − 2 Q = −1 e µ τ u c t Q = + 3 1 Q = 0 νe νµ ντ d s b Q = − 3 But for quarks, we have an additional mixing between families: u c t d s b This means that when emitting a W ±, an u quark for example will mostly change into a d quark, but it has a small chance to change into a s quark instead, and an even smaller chance to change into a b quark. Similarly, a c will mostly change into a s quark, but has small chances of changing into an u or b. Note that there is no horizontal mixing, i.e. an u never changes into a c quark! In practice, we will limit ourselves to the light quarks (u; d; s): 1 DRAWING FEYNMAN DIAGRAMS 2 u d s Some examples for diagrams emitting a W ±: W − W + e− νe u d And using quark mixing: W + u s To know the sign of the W -boson, we use charge conservation: the sum of the charges at the left hand side must equal the sum of the charges at the right hand side.
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